Question

The length \(x\) of a rectangle is decreasing at the rate of \(5 cm/minute\) and the width \(y\) is increasing at the rate of \(4 cm/minute\). When \(x = 8 cm\) and \(y = 6 cm\), find the rates of change of \((a)\) the perimeter, and \((b)\) the area of the rectangle.

Answer 1

The correct answer is \(2 cm^2/min.\)
Since the length \((x)\) is decreasing at the rate of \(5 cm/minute\) and the width \((y)\) is increasing at the rate of \(4 cm/minute\), we have:
\(\frac{dx}{dt}=-5cm/min\) and \(\frac{dy}{dt}=4 cm/min\)
(a) The perimeter \((P)\) of a rectangle is given by, \(P = 2(x + y)\)
\(∴ \frac{dp}{dt}=2(\frac{dx}{dt}+\frac{dy}{dt})=2(-5+4)=-2 cm/min.\)
(b) The area \((A)\) of a rectangle is given by, \(A = x \times y\)
\(∴\frac{dA}{dt}=\frac{dx}{dt}.y+x.\frac{dy}{dt}=-5y+4x\)
When \(x = 8 cm\) and \(y = 6 cm, \frac{dA}{dt}=(-5\times6+4\times8)cm^2/min=2cm^2/min\)
Hence, the area of the rectangle is increasing at the rate of \(2 cm^2/min.\)